Abstract
This study is an attempt to empirically detect the public opinion concerning majoritarian approval axiom. A social choice rule respects majoritarian approval iff it chooses only those alternatives which are regarded by a majority of “voters” to be among the “better half” of the candidates available. We focus on three social choice rules, the Majoritarian Compromise, Borda’s Rule and Condorcet’s Method, among which the Majoritarian Compromise is the only social choice rule always respecting majoritarian approval. We confronted each of our 288 subjects with four hypothetical preference profiles of a hypothetical electorate over some abstract set of four alternatives. At each hypothetical preference profile, two representing the preferences of five and two other of seven voters, the subject was asked to indicate, from an impartial viewpoint, which of the four alternatives should be chosen whose preference profile was presented, which if that is unavailable, then which if both of the above are unavailable, and finally which alternative should be avoided especially. In each of these profiles there is a Majoritarian Compromise-winner, a Borda-winner and a Condorcet-winner, and the Majoritarian Compromise-winner is always distinct from both the Borda-winner and the Condorcet-winner, while the Borda- and Condorcet-winners sometimes coincide. If the Borda- and Condorcet-winners coincide then there are two dummy candidates, otherwise only one, and dummies coincide with neither of the Majoritarian Compromise-, Borda- or Condorcet-winner. We presented our subjects with various types of hypothetical preference profiles, some where Borda respecting majoritarian approval, some where it failed to do so, then again for Condorcet, some profiles it respected majoritarian approval and some where it did not. The main thing we wanted to see was whether subjects’ support for Borda and Condorcet was higher when this social choice rule respected majoritarian approval than it did not. Our unambiguous overall empirical finding is that our subjects’ support for Borda and Condorcet was significantly stronger as they respect majoritarian approval.
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Notes
The joint winner of B and C will sometimes be denoted as B for Type I and Type II profiles in the rest of the paper.
This panel study obtains experimental data. That is, it is not a field research sampling from the “men in the street”. Hence, here we are not concerned with the representativeness of the sample.
It should be noted that the mean and median time spent by the subjects to answer the questionnaires is close to the ones in Sertel and Kara (2003). The extended time spent by the subjects can be explained by the observation that the students in Turkey are not used to get paid in return for answering questionnaires and when they are paid they feel obliged to seriously think before they answer them.
Business Administration, Chemical Engineering, Civil Engineering, Computer Engineering, Economics, History, International Finance, Mechanical Engineering, Media and Broadcasting, Physics, Politics.
During the pilot studies in Kara (2001), it is observed that the subjects who had taken a course of social choice theory fail to answer the questionnaires independent of what they had ‘learned’. That is, they treat the task as if it is a test of what they ‘know’ about the matter of concern. Apparently, this creates a bias in terms of impartiality. The students who had previously participated in a similar experiment were prevented to be the subjects of the present study since the subjects usually have ‘post-talks’ and get exposed the way the other subjects approach to the matter which also leads to a bias.
In all types of profile except for Type I, there are two “majoritarian approved” alternatives. One of these is always M-winner, the other is either B- or C- or B=C-winner.
Let \(\Delta _{i} = {\sum\nolimits_{,j \in {\left\{ {{\text{1,}}...{\text{,4}}} \right\}}} {{\left( {i{\text{ - }}j} \right)}q_{{ij}} } }\) for all i. Note that \(\Delta = {\sum\nolimits_{i \in {\left\{ {{\text{1,}}...{\text{,4}}} \right\}}} {\Delta _{i} } }\).
Let \(\Delta ^{j} = {\sum\nolimits_{i \in {\left\{ {{\text{1,}}...{\text{,4}}} \right\}}} {{\left( {i{\text{ - }}j} \right)}q_{{ij}} } }\) for all j. Note that \(\Delta = {\sum\nolimits_{j \in {\left\{ {{\text{1,}}...{\text{,4}}} \right\}}} {\Delta ^{j} } }\).
Below are the B-scores of the M, the B-, the C-winners and X at each root profile of Type IV:
Root profile
M
B
C
X
1
12
13
12
5
2
11
13
12
6
3
12
13
12
5
4
12
13
12
5
5
11
13
12
6
6
10
13
12
7
7
10
13
12
7
8
11
13
12
6
Of the 40 subjects second-ranking the C-winner, 24 were confronted with roots 2,5,6,7 and 8, and 26 were confronted with roots 1,3 and 4. The 24 subjects second-ranked the C-winner as a straightforward result of B-scoring. The 16 subjects, however, made their choice in favor of the C-winner at the profiles where the M- and the C-winners obtained the same B-score. Of the 28 subjects third-ranking the C-winner, 25 were confronted with root profiles of 1,3 and 4 and favored the C-winner and three others miscalculated the B-scores of the C-winner in roots 2,5 and 8
B-scores of the M-, the B-, the C-winners and X in the root profiles of Type II are as follows:
Root profile
M
B
C
X
1
12
13
12
5
2
11
13
12
6
3
11
13
12
6
4
10
13
12
7
Of the 49 subjects second-ranking the C-winner, 44 were confronted with roots 2,3, and 4, and 5 were confronted with root 1. The 44 subjects second-ranked the C-winner as a straightforward result of the ranking via B-scoring. The five subjects, however, made their choices in favor of the C-winner at root profile 1 where the M- and the C-winners obtain the same B-scores. Of the 19 subjects third-ranking the C-winner, eight were confronted with root 1 and favored the M-winner more than the C-winner and one subject was confronted with root 3, but miscalculated the B-scores and third-ranked the C-winner.
Since there is more than one subject with the same d i, an average rank is assigned to each of them.
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Acknowledgements
We would like to express our special thanks to Matthew Jackson and Charles Plott for the discussions through which our view are further refined. This article has also benefited from comments by two anonymous referees. The support of Turkish Academy of Sciences, and The Scientific and Technological Research Council of Turkey is gratefully acknowledged.
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This is one of the last published works of Murat R. Sertel whom we lost on January 25, 2003. He was a genuine scholar and an excellent teacher.His brilliant approach toward unifying the diversified topics of economic theory has deeply influenced his colleagues and students.
Appendices
Root profiles of type II
1. | ||||
B | B | X | X | Y |
M | M | B | M | M |
X | X | M | B | B |
Y | Y | Y | Y | X |
2. | ||||
B | B | X | X | Y |
M | M | B | M | M |
X | X | Y | B | B |
Y | Y | M | Y | X |
3. | ||||
B | B | X | X | Y |
M | M | B | M | M |
X | Y | Y | B | B |
Y | X | M | Y | X |
4. | ||||
B | B | X | X | Y |
M | M | B | M | M |
X | Y | M | B | B |
Y | X | Y | Y | X |
5. | ||||
B | B | X | X | Y |
M | M | B | M | M |
Y | Y | M | B | B |
X | X | Y | Y | X |
6. | ||||
B | B | X | X | Y |
M | M | B | M | M |
Y | Y | Y | B | B |
X | X | M | Y | X |
7. | ||||
B | B | X | X | Y |
M | M | B | M | M |
X | Y | Y | Y | B |
Y | X | M | B | X |
8. | ||||
B | B | X | X | Y |
M | M | B | M | M |
Y | Y | Y | Y | B |
X | X | M | B | X |
9. | ||||
B | B | X | X | Y |
M | M | B | M | M |
X | X | Y | Y | B |
Y | Y | M | B | X |
10. | ||||
B | B | X | X | Y |
M | M | M | M | B |
X | Y | B | B | M |
Y | X | Y | Y | X |
11. | ||||
B | B | X | X | Y |
M | M | M | M | B |
Y | Y | B | B | M |
X | X | Y | Y | X |
12. | ||||
B | B | X | X | Y |
M | M | M | M | B |
X | X | B | B | M |
Y | Y | Y | Y | X |
13. | ||||
B | B | X | X | Y |
M | M | M | M | B |
X | X | B | B | X |
Y | Y | Y | Y | M |
14. | ||||
B | B | X | X | Y |
M | M | M | M | B |
X | Y | B | B | X |
Y | X | Y | Y | M |
15. | ||||
B | B | X | X | Y |
M | M | M | M | B |
Y | Y | B | B | X |
X | X | Y | Y | M |
16. | ||||
B | B | X | X | Y |
M | M | M | M | B |
Y | Y | B | Y | X |
X | X | Y | B | M |
17. | ||||
B | B | X | X | Y |
M | M | M | M | B |
Y | X | B | Y | X |
X | Y | Y | B | M |
18. | ||||
B | B | M | X | X |
M | M | B | M | Y |
X | X | X | B | B |
Y | Y | Y | Y | M |
19. | ||||
B | B | M | X | X |
M | M | B | M | Y |
X | X | Y | B | B |
Y | Y | X | Y | M |
20. | ||||
B | B | M | X | X |
M | M | B | M | Y |
X | Y | Y | B | B |
Y | X | X | Y | M |
21. | ||||
B | B | M | X | X |
M | M | B | M | Y |
X | Y | X | B | B |
Y | X | Y | Y | M |
22. | ||||
B | B | M | X | X |
M | M | B | M | Y |
Y | Y | X | B | B |
X | X | Y | Y | M |
23. | ||||
B | B | M | X | X |
M | M | B | M | Y |
Y | Y | Y | B | B |
X | X | X | Y | M |
24. | ||||
B | B | M | X | X |
M | M | Y | B | M |
X | X | B | Y | B |
Y | Y | X | M | Y |
25. | ||||
B | B | M | X | X |
M | M | Y | B | M |
X | Y | B | Y | B |
Y | X | X | M | Y |
26. | ||||
B | B | M | X | X |
M | M | Y | B | M |
Y | Y | B | Y | B |
X | X | X | M | Y |
27. | ||||
B | B | M | X | Y |
M | M | B | M | X |
X | X | X | B | B |
Y | Y | Y | Y | M |
28. | ||||
B | B | M | X | Y |
M | M | B | M | X |
X | X | Y | B | B |
Y | Y | X | Y | M |
29. | ||||
B | B | M | X | Y |
M | M | B | M | X |
X | Y | X | B | B |
Y | X | Y | Y | M |
30. | ||||
B | B | M | X | Y |
M | M | B | M | X |
X | Y | Y | B | B |
Y | X | X | Y | M |
31. | ||||
B | B | M | X | Y |
M | M | B | M | X |
Y | Y | Y | B | B |
X | X | X | Y | M |
32. | ||||
B | B | M | X | Y |
M | M | B | M | X |
Y | Y | X | B | B |
X | X | Y | Y | M |
33. | ||||
B | B | M | X | Y |
M | M | X | B | M |
X | X | B | Y | B |
Y | Y | Y | M | X |
34. | ||||
B | B | M | X | Y |
M | M | X | B | M |
X | Y | B | Y | B |
Y | X | Y | M | X |
35. | ||||
B | B | M | X | Y |
M | M | X | B | M |
Y | Y | B | Y | B |
X | X | Y | M | X |
36. | ||||
B | B | M | X | Y |
M | M | Y | B | M |
X | X | B | Y | B |
Y | Y | X | M | X |
37. | ||||
B | B | M | X | Y |
M | M | Y | B | M |
X | Y | B | Y | B |
Y | X | X | M | X |
38. | ||||
B | B | M | X | Y |
M | M | Y | B | M |
Y | Y | B | Y | B |
X | X | X | M | X |
An exemplary questionnaire
University:
Department:
Year:
A group of citizens faces four alternatives. Exactly one of these is to be adopted.
Each citizen ranks the four alternatives according to his/her own preference. For example, a member ranking the alternatives as
-
a
-
b
-
c
-
d
has ranked “a” as his/her top choice, “b” as his/her second choice, “c” as his/her third choice and “d” as his/her last choice.
Below are presented four distinct groups whose members (citizens) exhibit various rankings of the alternatives according to their personal preferences. For each group, taking an impartial point of view, you are asked to indicate which alternative (“a” or “b” or “c” or “d”) should be adopted, which should be adopted if this becomes unavailable, and which should be adopted that if, too, becomes unavailable, and which should especially be avoided. You are also encouraged to give a brief explanation concerning the reasoning on which your views rest for each of the four groups.
1.mbr. | 2.mbr. | 3.mbr. | 4.mbr. | 5.mbr. | 6.mbr. | 7.mbr. |
a | b | c | a | c | a | c |
b | d | b | c | d | b | b |
c | a | a | b | a | c | d |
d | c | d | d | b | d | a |
1.mbr. | 2.mbr. | 3.mbr. | 4.mbr. | 5.mbr. | 6.mbr. | 7.mbr. |
d | a | b | b | d | b | d |
c | d | c | a | c | c | c |
b | b | a | c | b | d | b |
a | c | d | d | a | a | a |
1.mbr. | 2.mbr. | 3.mbr. | 4.mbr. | 5.mbr. | ||
a | c | b | b | a | ||
d | d | d | c | d | ||
b | a | a | a | b | ||
c | b | c | d | c |
1.mbr. | 2.mbr. | 3.mbr. | 4.mbr. | 5.mbr. | ||
a | d | a | d | b | ||
d | c | c | c | c | ||
c | b | d | b | d | ||
b | a | b | a | a |
Self-selectivity results by root profiles
* Type I (M/B∼C)
| Selecting SCR | Selected SCR |
---|---|---|
Root profile 1 | M | B=C |
B | B=C | |
C | B=C | |
Root profile 2 | M | B=C |
B | B=C | |
C | B=C | |
Root profile 3 | M | B=C |
B | B=C | |
C | B=C |
* Type II (M×B∼C/)
| Selecting SCR | Selected SCR |
---|---|---|
Root profile 1 | M | B=C |
B | B=C | |
C | B=C | |
Root profile 2 | M | B=C |
B | B=C | |
C | B=C | |
Root profile 3 | M | B=C |
B | B=C | |
C | B=C |
* Type III (M×B/C)
| Selecting SCR | Selected SCR |
---|---|---|
Root profile 1 | M | B |
B | B | |
C | B | |
Root profile 2 | M | B |
B | B | |
C | B | |
Root profile 3 | M | B |
B | B | |
C | B | |
Root profile 4 | M | B |
B | B | |
C | B |
* Type IV (M×C/B)
| Selecting SCR | Selected SCR |
---|---|---|
Root profile 1 | M | M |
B | M | |
C | M | |
Root profile 2 | M | C |
B | C | |
C | C | |
Root profile 3 | M | B |
B | B | |
C | B | |
Root profile 4 | M | M |
B | B and M | |
C | No winner | |
Root profile 5 | M | C |
B | C | |
C | C | |
Root profile 6 | M | C |
B | C | |
C | No winner | |
Root profile 7 | M | C |
B | B | |
C | No winner | |
Root profile 8 | M | B |
B | B | |
C | B |
Majoritarian approval results by root profiles
* Type I (M/B∼C)
| M (%) | B=C (%) |
---|---|---|
Root profile 1 | 72.9 | 80.2 |
Root profile 2 | 89.6 | 80.2 |
Root profile 3 | 85.4 | 82.3 |
* Type II (M×B∼C/)
| M (%) | B=C (%) |
---|---|---|
Root profile 1 | 79.2 | 96.9 |
Root profile 2 | 99.0 | 98.0 |
Root profile 3 | 66.6 | 90.6 |
* Type III (M×B/C)
| M (%) | B (%) | C (%) |
---|---|---|---|
Root profile 1 | 45.8 | 91.7 | 61.1 |
Root profile 2 | 45.8 | 94.4 | 58.3 |
Root profile 3 | 48.6 | 89.9 | 61.1 |
Root profile 4 | 48.6 | 91.7 | 59.8 |
* Type IV (M×C/B)
| M (%) | B (%) | C (%) |
---|---|---|---|
Root profile 1 | 80.5 | 63.9 | 52.8 |
Root profile 2 | 50.0 | 61.1 | 88.9 |
Root profile 3 | 88.9 | 61.1 | 49.0 |
Root profile 4 | 80.6 | 63.9 | 55.6 |
Root profile 5 | 58.3 | 58.3 | 83.4 |
Root profile 6 | 61.1 | 63.9 | 75.0 |
Root profile 7 | 66.6 | 58.3 | 75.0 |
Root profile 8 | 47.2 | 86.1 | 66.7 |
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Giritligil Kara, A.E., Sertel, M.R. Does majoritarian approval matter in selecting a social choice rule? An exploratory panel study. Soc Choice Welfare 25, 43–73 (2005). https://doi.org/10.1007/s00355-005-0024-8
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DOI: https://doi.org/10.1007/s00355-005-0024-8